相关论文: $k$-Schur expansions of Catalan functions

We study Type C $K$-Stanley symmetric functions, which are $K$-theoretic extensions of the Type C Stanley symmetric functions. They are indexed by signed permutations and can be used to enumerate reduced words via their expansion into Schur…

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum…

We prove Okounkov's conjecture, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. An…

We introduce two families of symmetric functions generalizing the factorial Schur $P$- and $Q$- functions due to Ivanov. We call them $K$-theoretic analogues of factorial Schur $P$- and $Q$- functions. We prove various combinatorial…

We prove the Murgnaghan--Nakayama rule for $k$-Schur functions of Lapointe and Morse, that is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a $k$-Schur function in terms of $k$-Schur…

We formulate many open questions regarding the Schur positivity of the effect of interesting operators on symmetric functions, and give supporting evidence for why one should expect such behavior.

In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula is general enough to give new expressions for all known types of Schubert polynomials. In the present paper we…

Multi-Schur functions are symmetric functions that generalize the supersymmetric Schur functions, the flagged Schur functions, and the refined dual Grothendieck functions, which have been intensively studied by Lascoux. In this paper, we…

Rado's theorem about permutahedra and dominance order on partitions reveals that each Schur polynomial is M-convex, or equivalently, it has a saturated Newton polytope and this polytope is a generalized permutahedron as well. In this paper…

Macdonald defined two-parameter Kostka functions K_{\lambda\mu}(q,t) where \lambda, \mu are partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture…

Recently we explained that the classical $Q$ Schur functions stand behind various well-known properties of the cubic Kontsevich model, and the next step is to ask what happens in this approach to the generalized Kontsevich model (GKM) with…

We show that Lapointe-Lascoux-Morse k-Schur functions (at t=1) and Fomin-Gelfand-Postnikov quantum Schubert polynomials can be obtained from each other by a rational substitution. This is based upon Kostant's solution of the Toda lattice…

We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{\mu\le\lambda} g^{(k)}_{\mu}$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by…

Haglund, Morse, and Zabrocki introduced a family of creation operators of Hall-Littlewood polynomials, $\{C_{a}\}$ for any $a\in \mathbb{Z}$, in their compositional refinement of the shuffle (ex-)conjecture. For any $\alpha\vDash n$, the…

We present a family of analogs of the Hall-Littlewood symmetric functions in the $Q$-function algebra. The change of basis coefficients between this family and Schur's $Q$-functions are $q$-analogs of numbers of marked shifted tableaux.…

This paper gives a tableau formula for expanding the product of a Lascoux polynomial and a stable Grothendieck polynomial into Lascoux polynomials. Lascoux and stable Grothendieck polynomials are inhomogeneous analogues of key polynomials…

We classify the $Q$-homogeneous skew Schur $Q$-functions, i.e., those of the form $Q_{\lambda/\mu} = k \cdot Q_{\nu}$. On the way we develop new tools that are useful also in the context of other classification problems for skew Schur…

MacMahon's classic generating function of random plane partitions, which is related to Schur polynomials, was recently extended by Vuletic to a generating function of weighted plane partitions that is related to Hall-Littlewood polynomials,…

This paper is about a family of symmetric rational functions that form a one-parameter generalization of the classical Hall-Littlewood polynomials. We introduce two sets of (skew and non-skew) functions that are akin to P and Q…

The branching of (k-1)-Schur functions into k-Schur functions was given by Lapointe, Lam, Morse and Shimozono as chains in a poset on k-shapes. The k-Schur functions are the parameterless case of a more general family of symmetric functions…